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									            The Pigeonhole Principle




9/18/2012                              1
 The Pigeonhole Principle

• In words:
    – If n pigeons are in
      fewer than n
      pigeonholes, some
                            n
      pigeonhole must
      contain at least
      two pigeons

                                What is n?
                                         http://www.blog.republicofmath.com/archives/3115




                                                                                2
 9/18/2012
  The Pigeonhole Principle

  • In math:
Let f : A  B, where A and B
are finite sets and A  B .
Then there exist distinct elements
a1 ,a2 A such that f (a1 )  f (a2 ).




                                         3
  9/18/2012
The Pigeonhole Principle
                                 Let f : A  B, where A and B
 •   What is a set?
                         are finite sets and A  B .
 •   a finite set?       Then there exist distinct elements
 •   What is |A|?        a1 ,a2 A such that f (a1 )  f (a2 ).

 •   What is a function?
 •   the domain of a function?
 •   the codomain of a function?
 •   Why say “distinct”?

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9/18/2012
Applications of The Pigeonhole Principle

• In any group of 8 people, two were born
  on the same day of the week
• What are the “pigeons” and what are the
  “pigeonholes”?
• A = the set of people, B = {Sun, … Sat},
  f(a) = the day of the week on which a was
  born


                                           5
9/18/2012
   Applications of The Pigeonhole Principle

• Suppose each pigeonhole        D   D   D   D   D
  contains one bird, and         D   D   D   D   D
  every bird moves to an
                                 D   D   D   D   D
  adjacent square (up, down,
  left or right). Show that no   D   D   D   D   D
  matter how this is done,       D   D   D   D   D
  some pigeonhole winds up
  with at least 2 birds.



                                                 6
   9/18/2012
   Applications of The Pigeonhole Principle

• Suppose each pigeonhole        D   D   D   D   D
  contains one bird, and         D   D   D   D   D
  every bird moves to an
                                 D   D   D   D   D
  adjacent square (up, down,
  left or right). Show that no   D   D   D   D   D
  matter how this is done,       D   D   D   D   D
  some pigeonhole winds up
  with at least 2 birds.



                                                 7
   9/18/2012
   Applications of The Pigeonhole Principle

• Suppose each pigeonhole          D     D    D     D       D
  contains one bird, and           D     D    D     D       D
  every bird moves to an
                                   D     D    D     D       D
  adjacent square (up, down,
  left or right). Show that no     D     D    D     D       D
  matter how this is done,         D     D    D     D       D
  some pigeonhole winds up       A  birds on red squares
  with at least 2 birds.         B  gray squares
                                 f (a)  the square a moves to
                                 A  13, B  12

                                                            8
   9/18/2012

								
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